Determine all triples $(x,y,z)$ of integers satisfying the equation $3x+4y+5z=6$
I am not familiar with Diophantine equations with more variables. How do I solve this? Please anyone suggest some easier way to solve these linear Diophantine equations.
I think this can help here -
If $(x_0, y_0)$ are solutions to $ax + by = C$ with $gcd(a,b) = 1$, then $(x_0 + bt, y_0 - at)$ for all integer $t$ are also solutions.
Note that we have that $$3x+4y=6-5z$$ So from what you know, we have that $(x,y)$ is characterized by $(-2+z-4t,3-2z+3t)$ for some integer $t$. So $(x,y,z)$ are for $t,z \in \mathbb{Z}$, $$(-2+z-4t,3-2z+3t,z)$$